Subleading Logarithms Research Articles

Resummation of Next-to-Leading Nonglobal Logarithms at the LHC.

In cross sections with angular cuts, an intricate pattern of enhanced higher-order corrections known as nonglobal logarithms arises. The leading logarithmic terms were computed numerically two decades ago, but the resummation of subleading nonglobal logarithms remained a challenge that we solve in this Letter using renormalization group methods in effective field theory. To achieve next-to-leading logarithmic accuracy, we implement the two-loop anomalous dimension governing the resummation of nonglobal logarithms into a large-N_{c} parton shower framework, together with one-loop matching corrections. As a first application, we study the interjet energy flow in e^{+}e^{-} annihilation into two jets. We then present, for the first time, resummed predictions at next-to-leading logarithmic accuracy for a gap-between-jets observable at hadron colliders.

Two-loop anomalous dimension for the resummation of non-global observables

The soft radiation emitted in jet cross sections can resolve the directions and colors of individual hard partons, leading to a complicated pattern of logarithmically enhanced terms in the perturbative series. Starting from a factorization theorem and solving the renormalization group equations for its ingredients, these large logarithms can be resummed. In this paper we extract the two-loop anomalous dimension governing the resummation of subleading logarithms in jet cross sections and other non-global observables. This anomalous dimension can be obtained by considering soft limits of hard amplitudes, but the presence of collinear singularities in intermediate expressions makes its extraction delicate. As a consistency check, we use our results to predict the known subleading non-global logarithms in the two-jet cross section.

Exact result in mathcal{N} = 4 SYM theory: generalised double-logarithmic equation

We present the new results for the generalised double-logarithmic equation, obtained from the analytical continuation of the seven-loop anomalous dimension of twist-2 operators in the planar mathcal{N} = 4 SYM theory. The double-logarithmic equation is related to the special asymptotic of the scattering amplitudes, when the large logarithms of the energy of scattering particles are appeared and should be summed in all order of perturbative theory. These large logarithms correspond to the poles of the analytically continued anomalous dimension. The generalised double-logarithmic equation includes the subleading logarithms. We have found, that the expansion of the generalised double-logarithmic equation can be ressumed in the form of rational functions with simple denominator. The solution of the generalised double-logarithmic equation provides a lot of information about the poles of the analytically continued anomalous dimension in all orders of perturbative theory. We have found also the generalised double-logarithmic equation for the analytically continued anomalous dimension near the value, which is related with BFKL-equation.

The LPM effect in sequential bremsstrahlung: analytic results for sub-leading (single) logarithms

Consider the in-medium splitting g → gg of a very high-energy gluon traversing a QCD medium, accounting for the Landau-Pomeranchuk-Migdal (LPM) effect. It has been known for some time that soft radiative corrections to that splitting generate a double-log correction to the splitting rate, whose effects can be absorbed into running of the medium parameter hat{q} describing the rate of transverse momentum kicks to high-energy particles due to small-angle scattering from the medium. Less has been known about sub-leading, single logarithms in this context. In this paper, we find analytic formulas for those single logs (with various caveats and clarifications).

Combined subleading high-energy logarithms and NLO accuracy for W production in association with multiple jets

Large logarithmic corrections in hat{s}/{p}_t^2 lead to substantial variations in the perturbative predictions for inclusive W-plus-dijet processes at the Large Hadron Collider. This instability can be cured by summing the leading-logarithmic contributions in hat{s}/{p}_t^2 to all orders in αs. As expected though, leading logarithmic accuracy is insufficient to guarantee a suitable description in regions of phase space away from the high energy limit.We present (i) the first calculation of all partonic channels contributing at next-to-leading logarithmic order in W-boson production in association with at least two jets, and (ii) bin-by-bin matching to next-to-leading fixed-order accuracy. This new perturbative input is implemented in High Energy Jets, and systematically improves the description of available experimental data in regions of phase space which are formally subleading with respect to hat{s}/{p}_t^2 .

Factorization at subleading power and endpoint divergences in h \u2192 \u03b3\u03b3 decay. Part II. Renormalization and scale evolution

Building on the recent derivation of a bare factorization theorem for the b-quark induced contribution to the h → γγ decay amplitude based on soft-collinear effective theory, we derive the first renormalized factorization theorem for a process described at subleading power in scale ratios, where λ = mb/Mh « 1 in our case. We prove two refactorization conditions for a matching coefficient and an operator matrix element in the endpoint region, where they exhibit singularities giving rise to divergent convolution integrals. The refactorization conditions ensure that the dependence of the decay amplitude on the rapidity regulator, which regularizes the endpoint singularities, cancels out to all orders of perturbation theory. We establish the renormalized form of the factorization formula, proving that extra contributions arising from the fact that “endpoint regularization” does not commute with renormalization can be absorbed, to all orders, by a redefinition of one of the matching coefficients. We derive the renormalization-group evolution equation satisfied by all quantities in the factorization formula and use them to predict the large logarithms of order {alpha alpha}_s^2{L}^k in the three-loop decay amplitude, where L=ln left(-{M}_h^2/{m}_b^2right) and k = 6, 5, 4, 3. We find perfect agreement with existing numerical results for the amplitude and analytical results for the three-loop contributions involving a massless quark loop. On the other hand, we disagree with the results of previous attempts to predict the series of subleading logarithms sim {alpha alpha}_s^n{L}^{2n+1} .

Solution to the Sudakov suppressed Balitsky-Kovchegov equation and its application to HERA data * *Supported by the National Natural Science Foundation of China (11765005, 11305040, 11947119, 11847152), the Fund of Science and Technology Department of Guizhou Province ([2018]1023,[2019]5653), the Education Department of Guizhou Province (KY[2017]004), the National Key Research and Development Program of China (2018YFE0104700, CCNU18ZDPY04)

We analytically solve the Sudakov suppressed Balitsky-Kovchegov evolution equation with fixed and running coupling constants in the saturation region. The analytic solution of the S-matrix shows that the rapidity dependence of the solution with the fixed coupling constant is replaced by the dependence in the smallest dipole running coupling case, as opposed to obeying the law found in our previous publication, where all the solutions of the next-to-leading order evolution equations comply with rapidity dependence once the QCD coupling is switched from the fixed coupling to the smallest dipole running coupling prescription. This finding indicates that the corrections of the sub-leading double logarithms in the Sudakov suppressed evolution equation are significant, which compensate for a part of the evolution decrease of the dipole amplitude introduced by the running coupling effect. To test the analytic findings, we calculate the numerical solutions of the Sudakov suppressed evolution equation, and the numerical results confirm the analytic outcomes. Moreover, we use the numerical solutions of the evolution equationto fit the HERA data. This demonstrates that the Sudakov suppressed evolution equation can achieve a good quality fit to the data.

Subleading power resummation of rapidity logarithms: the energy-energy correlator in mathcal{N} = 4 SYM

We derive and solve renormalization group equations that allow for the resummation of subleading power rapidity logarithms. Our equations involve operator mixing into a new class of operators, which we term the “rapidity identity operators”, that will generically appear at subleading power in problems involving both rapidity and virtuality scales. To illustrate our formalism, we analytically solve these equations to resum the power suppressed logarithms appearing in the back-to-back (double light cone) limit of the Energy-Energy Correlator (EEC) in mathcal{N} = 4 super-Yang-Mills. These logarithms can also be extracted to mathcal{O}left({alpha}_s^3right) from a recent perturbative calculation, and we find perfect agreement to this order. Instead of the standard Sudakov exponential, our resummed result for the subleading power logarithms is expressed in terms of Dawson’s integral, with an argument related to the cusp anomalous dimension. We call this functional form “Dawson’s Sudakov”. Our formalism is widely applicable for the resummation of subleading power rapidity logarithms in other more phenomenologically relevant observables, such as the EEC in QCD, the pT spectrum for color singlet boson production at hadron colliders, and the resummation of power suppressed logarithms in the Regge limit.

Eccentric-orbit extreme-mass-ratio-inspiral radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders

We present new results on the analytic eccentricity dependence of several sequences of gravitational wave flux terms at high post-Newtonian (PN) order for extreme-mass-ratio inspirals. These sequences are the leading logarithms, which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ for integers $k\ge 0$ ($x$ a PN compactness parameter), and the subleading logarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2} \log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated to infinity. For the energy flux leading logarithms, we show that to arbitrarily high PN order their eccentricity dependence is determined by particular sums over the function $g(n,e_t)$, derived from the Newtonian mass quadrupole moment, that normally gives the spectral content of the Peters-Mathews flux as a function of radial harmonic $n$. An analogous power spectrum $\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentum flux. For subleading logs, the quadrupole power spectra are again shown to play a role, providing a distinguishable part of the eccentricity dependence of these flux terms to high PN order. With the quadrupole contribution understood, the remaining analytic eccentricity dependence of the subleading logs can in principle be determined more easily using black hole perturbation theory. We show this procedure in action, deriving the complete analytic structure of the $x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$ subleading log to high order in a power series in eccentricity. We discuss how these methods might be extended to other sequences of terms in the PN expansion involving logarithms.

Sudakov suppression of the Balitsky-Kovchegov kernel

To sum high-energy leading logarithms in a consistent way, one has to impose the strong ordering in both projectile rapidity and dense target rapidity simultaneously, which results in a kinematically improved Balitsky-Kovchegov (BK) equation. We find that beyond this strong ordering region, the important sub-leading double logarithms arise at high order due to the incomplete cancellation between real corrections and virtual corrections in a t-channel calculation. Based on this observation, we further argue that these double logarithms are the Sudakov type ones, and thus can be resummed into an exponential leading to a Sudakov suppressed BK equation.

Four-Gluon Scattering at Three Loops, Infrared Structure, and the Regge Limit.

We compute the three-loop four-gluon scattering amplitude in maximally supersymmetric Yang-Mills theory, including its full color dependence. Our result is the first complete computation of a nonplanar four-particle scattering amplitude to three loops in four-dimensional gauge theory and consequently provides highly nontrivial data for the study of nonplanar scattering amplitudes. We present the amplitude as a Laurent expansion in the dimensional regulator to finite order, with coefficients composed of harmonic polylogarithms of uniform transcendental weight, and simple rational prefactors. Our computation provides an independent check of a recent result for three-loop corrections to the soft anomalous dimension matrix that predicts the general infrared singularity structure of massless gauge theory scattering amplitudes. Taking the Regge limit of our result, we determine the three-loop gluon Regge trajectory. We also find agreement with very recent predictions for subleading logarithms.

Transverse momentum resummation for dijet correlation in hadronic collisions

We study the transverse momentum resummation for dijet correlation in hadron collisions based on the Collins-Soper-Sterman formalism. The complete one-loop calculations are carried out in the collinear factorization framework for the differential cross sections at low imbalance transverse momentum between the two jets. Important cross checks are performed to demonstrate that the soft divergences cancelled out between different diagrams, and in particular, those associated with final state jets. The leading and sub-leading logarithms are identified. All order resummation is derived following the transverse momentum dependent factorization at this order. Its phenomenological applications are also presented.

Precise Higgs-background predictions: merging NLO QCD and squared quark-loop corrections to four-lepton + 0,1 jet production

We present precise predictions for four-lepton plus jets production at the LHC obtained within the fully automated Sherpa+OpenLoops framework. Off-shell intermediate vector bosons and related interferences are consistently included using the complex-mass scheme. Four-lepton plus 0- and 1-jet final states are described at NLO accuracy, and the precision of the simulation is further increased by squared quark-loop NNLO contributions in the gg -> 4l, gg -> 4l+g, gq -> 4l+q, and qq -> 4l+g channels. These NLO and NNLO contributions are matched to the Sherpa parton shower, and the 0- and 1-jet final states are consistently merged using the MEPS@NLO technique. Thanks to Sudakov resummation, the parton shower provides improved predictions and uncertainty estimates for exclusive observables. This is important when jet vetoes or jet bins are used to separate four-lepton final states arising from Higgs decays, diboson production, and top-pair production. Detailed predictions are presented for the ATLAS and CMS H->WW analyses at 8 TeV in the 0- and 1-jet bins. Assessing renormalisation-, factorisation- and resummation-scale uncertainties, which reflect also unknown subleading Sudakov logarithms in jet bins, we find that residual perturbative uncertainties are as small as a few percent.

Finite-size scaling method for the Berezinskii–Kosterlitz–Thouless transition

We test an improved finite-size scaling method for reliably extracting the critical temperature TBKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρs at TBKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρs(TBKT) = 2TBKT/π, we define a size-dependent transition temperature TBKT(L1,L2) based on a pair of system sizes L1,L2, e.g., L2 = 2L1. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining TBKT(L1,L2). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result TBKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, TBKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated TBKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.

Collinear and soft-gluon corrections to Higgs production at next-to-next-to-next-to-leading order

I present analytical expressions for the collinear and soft-gluon corrections to Higgs production via the process $b\overline{b}\ensuremath{\rightarrow}H$ as well as $gg\ensuremath{\rightarrow}H$ through next-to-next-to-next-to-leading order (NNNLO). The soft corrections are complete, while the collinear corrections include leading and some subleading logarithms. Numerical results at the Tevatron and the CERN LHC are presented, primarily for $b\overline{b}\ensuremath{\rightarrow}H$. It is shown that the collinear terms greatly improve the soft and virtual approximation at next-to-leading order and next-to-next-to-leading order, especially when subleading terms are included. The NNNLO collinear and soft corrections provide significant enhancements to the total cross section. I also provide expressions for the collinear and soft corrections through NNNLO for the related Drell-Yan process.

Renormalization group determination of the five-loop effective potential for massless scalar field theory

The five-loop effective potential and the associated summation of subleading logarithms for O(4) globally symmetric massless λϕ4 field theory in the Coleman–Weinberg renormalization scheme d4V / dϕ4|ϕ=μ = λ (where μ is the renormalization scale) is calculated via renormalization-group methods. An important aspect of this analysis is conversion of the known five-loop renormalization-group functions in the minimal-subtraction (MS) scheme to the Coleman–Weinberg scheme.PACS Nos.: 11.30.Qc, 11.10.Hi, 11.15.Tk, 12.15.Lk

Threshold resummation for Higgs production in effective field theory

We present an effective field theory to resum the large double logarithms originated from soft-gluon radiations at small final-state hadron invariant masses in Higgs and vector boson (\gamma^*, $W$ and $Z$) production at hadron colliders. The approach is conceptually simple, indepaendent of details of an effective field theory formulation, and valid to all orders in sub-leading logarithms. As an example, we show the result of summing the next-to-next-to-next leading logarithms is identical to that of standard pQCD factorization method.

Resummation of large logarithms within the method of effective charges

We show how the resummation of large logarithms can be incorporated into the method of effective charges. As an example, we apply this approach to the event shape variables thrust and heavy-jet mass in e + e − annihilation. We find that, although the resummation creates problems with the behaviour of the effective charge in the 2-jet limit, smaller power corrections are required to fit the data compared to the standard approach. In addition, increasing the logarithmic accuracy reduces the size of the power corrections further. We also consider “predicting” the sub-leading logarithms in the MS ¯ scheme, obtaining surprisingly good results for the first few NLL terms.

SUDAKOV EXPANSIONS AT ONE-LOOP AND BEYOND FOR CHARGED SCALAR AND FERMION PAIR PRODUCTION IN SUSY MODELS AT FUTURE LINEAR COLLIDERS

We consider the high energy behavior of the amplitudes for pair production of charged leptons, quarks, Higgs bosons, sleptons, squarks and charginos at lepton colliders. We give the general expressions of the leading quadratic and subleading linear logarithms that appear at the one-loop level, and derive the corresponding resummed expansions to subleading logarithmic order accuracy. Under the assumption of a relatively light SUSY scenario and choosing the MSSM as a specific model, we compare the predictions of the one-loop and of the resummed expansions at variable energy. We show that the two predictions are very close in the one TeV regime, but drastically differ in the few (2,3) TeV range.

Fermionic and scalar corrections for the Abelian form factor at two loops

Two-loop corrections for the form factor in a massive Abelian theory are evaluated, which result from the insertion of massless fermion or scalar loops into the massive gauge boson propagator. The result is valid for arbitrary energies and gauge boson mass. Power-suppressed terms vanish rapidly in the high-energy region where the result is well approximated by a polynomial of third order in ln(s/M2). The relative importance of subleading logarithms is emphasized.